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• Chapter 1

Delta-ring

In mathematics, a nonempty collection of setsR is called a -ring (pronounced delta-ring) if it is closed under union,relative complementation, and countable intersection:

1. A [B 2 R if A;B 2 R2. AB 2 R if A;B 2 R3. T1n=1An 2 R if An 2 R for all n 2 N

If only the rst two properties are satised, then R is a ring but not a -ring. Every -ring is a -ring, but not every-ring is a -ring.-rings can be used instead of -elds in the development of measure theory if one does not wish to allow sets ofinnite measure.

1.2 References Cortzen, Allan. Delta-Ring. From MathWorldA Wolfram Web Resource, created by Eric W. Weisstein.http://mathworld.wolfram.com/Delta-Ring.html

2

• Chapter 2

Field of sets

Set algebra redirects here. For the basic properties and laws of sets, see Algebra of sets.

In mathematics a eld of sets is a pair hX;Fi whereX is a set andF is an algebra over X i.e., a non-empty subsetof the power set of X closed under the intersection and union of pairs of sets and under complements of individualsets. In other words F forms a subalgebra of the power set Boolean algebra of X . (Many authors refer to F itselfas a eld of sets. The word eld in eld of sets is not used with the meaning of eld from eld theory.) Elementsof X are called points and those of F are called complexes and are said to be the admissible sets of X .Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can berepresented as a eld of sets.

2.1 Fields of sets in the representation theory of Boolean algebras

2.1.1 Stone representationEvery nite Boolean algebra can be represented as a whole power set - the power set of its set of atoms; each elementof the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). This power setrepresentation can be constructed more generally for any complete atomic Boolean algebra.In the case of Boolean algebras which are not complete and atomic we can still generalize the power set representationby considering elds of sets instead of whole power sets. To do this we rst observe that the atoms of a nite Booleanalgebra correspond to its ultralters and that an atom is below an element of a nite Boolean algebra if and only ifthat element is contained in the ultralter corresponding to the atom. This leads us to construct a representation of aBoolean algebra by taking its set of ultralters and forming complexes by associating with each element of the Booleanalgebra the set of ultralters containing that element. This construction does indeed produce a representation of theBoolean algebra as a eld of sets and is known as the Stone representation. It is the basis of Stones representationtheorem for Boolean algebras and an example of a completion procedure in order theory based on ideals or lters,similar to Dedekind cuts.Alternatively one can consider the set of homomorphisms onto the two element Boolean algebra and form complexesby associating each element of the Boolean algebra with the set of such homomorphisms that map it to the topelement. (The approach is equivalent as the ultralters of a Boolean algebra are precisely the pre-images of the topelements under these homomorphisms.) With this approach one sees that Stone representation can also be regardedas a generalization of the representation of nite Boolean algebras by truth tables.

2.1.2 Separative and compact elds of sets: towards Stone duality A eld of sets is called separative (or dierentiated) if and only if for every pair of distinct points there is acomplex containing one and not the other.

Aeld of sets is called compact if and only if for every proper lter overX the intersection of all the complexes

3

• 4 CHAPTER 2. FIELD OF SETS

contained in the lter is non-empty.

These denitions arise from considering the topology generated by the complexes of a eld of sets. Given a eld ofsets X = hX;Fi the complexes form a base for a topology, we denote the corresponding topological space by T (X). Then

T (X) is always a zero-dimensional space. T (X) is a Hausdor space if and only if X is separative. T (X) is a compact space with compact open sets F if and only if X is compact. T (X) is a Boolean space with clopen sets F if and only if X is both separative and compact (in which case itis described as being descriptive)

The Stone representation of a Boolean algebra is always separative and compact; the corresponding Boolean space isknown as the Stone space of the Boolean algebra. The clopen sets of the Stone space are then precisely the complexesof the Stone representation. The area of mathematics known as Stone duality is founded on the fact that the Stonerepresentation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a dualityexists between Boolean algebras and Boolean spaces.

2.2 Fields of sets with additional structure

2.2.1 Sigma algebras and measure spacesIf an algebra over a set is closed under countable intersections and countable unions, it is called a sigma algebraand the corresponding eld of sets is called a measurable space. The complexes of a measurable space are calledmeasurable sets.A measure space is a triple hX;F ; i where hX;Fi is a measurable space and is a measure dened on it. If is in fact a probability measure we speak of a probability space and call its underlying measurable space a samplespace. The points of a sample space are called samples and represent potential outcomes while the measurable sets(complexes) are called events and represent properties of outcomes for which we wish to assign probabilities. (Manyuse the term sample space simply for the underlying set of a probability space, particularly in the case where everysubset is an event.) Measure spaces and probability spaces play a foundational role in measure theory and probabilitytheory respectively.The Loomis-Sikorski theorem provides a Stone-type duality between abstract sigma algebras and measurable spaces.

2.2.2 Topological elds of setsA topological eld of sets is a triple hX; T ;Fi where hX; T i is a topological space and hX;Fi is a eld of setswhich is closed under the closure operator of T or equivalently under the interior operator i.e. the closure and interiorof every complex is also a complex. In other words F forms a subalgebra of the power set interior algebra on hX; T i.Every interior algebra can be represented as a topological eld of sets with its interior and closure operators corre-sponding to those of the topological space.Given a topological space the clopen sets trivially form a topological eld of sets as each clopen set is its own interiorand closure. The Stone representation of a Boolean algebra can be regarded as such a topological eld of sets.

Algebraic elds of sets and Stone elds

A topological eld of sets is called algebraic if and only if there is a base for its topology consisting of complexes.If a topological eld of sets is both compact and algebraic then its topology is compact and its compact open sets areprecisely the open complexes. Moreover the open complexes form a base for the topology.

• 2.2. FIELDS OF SETS WITH ADDITIONAL STRUCTURE 5

Topological elds of sets that are separative, compact and algebraic are calledStone elds and provide a generalizationof the Stone representation of Boolean algebras. Given an interior algebra we can form the Stone representation ofits underlying Boolean algebra and then extend this to a topological eld of sets by taking the topology generated bythe complexes corresponding to the open elements of the interior algebra (which form a base for a topology). Thesecomplexes are then precisely the open complexes and the construction produces a Stone eld representing the interioralgebra - the Stone representation.

2.2.3 Preorder eldsA preorder eld is a triple hX;;Fi where hX;i is a preordered set and hX;Fi is a eld of sets.Like the topological elds of sets, preorder elds play an important role in the representation theory of interior alge-bras. Every interior algebra can be represented as a preorder eld with its interior and closure operators correspondingto those of the Alexandrov topology induced by the preorder. In other words

Int(S) = fx 2 X : there exists a y 2 S with y xg andCl(S) = fx 2 X : there exists a y 2 S with x yg for all S 2 F

Preorder elds arise naturally in modal logic where the points represent the possible worlds in the Kripke semanticsof a theory in the modal logic S4 (a formal mathematical abstraction of epistemic logic), the preorder represents theaccessibility relation on these possible worlds in this semantics, and the complexes represent sets of possible worldsin which individual sentences in the theory hold, providing a representation of the Lindenbaum-Tarski algebra of thetheory.

Algebraic and canonical preorder elds

A preorder eld is called algebraic if and only if it has a set of complexes A which determines the preorder in thefollowing manner: x y if and only if for every complex S 2 A , x 2 S implies y 2 S . The preorder eldsobtained from S4 theories are always algebraic, the complexes determining the preorder being the sets of possibleworlds in which the sentences of the theory closed under necessity hold.A separative compact algebraic preorder eld is said to be canonical. Given an interior algebra, by replacing thetopology of its Stone representation with the corresponding canonical preorder (specialization preorder) we obtain arepresentation of the interior algebra as a canonical preorder eld. By replacing the preorder by its correspondingAlexandrov topology we obtain an alternative representation of the interior algebra as a topological eld of sets. (Thetopology of this "Alexandrov representation" is just the Alexandrov bi-coreection of the topology of the Stonerepresentation.)

2.2.4 Complex algebras and elds of sets on relational structuresThe representation of interior algebras by preorder elds can be generalized to a representation theorem for arbi-trary (normal) Boolean algebras with operators. For this we consider structures hX; (Ri)I ;Fi where hX; (Ri)Ii is arelational structure i.e. a set with an indexed family of relations dened on it, and hX;Fi is a eld of sets. The com-plex algebra (or algebra of complexes) determined by a eld of sets X = hX; (Ri)I ;Fi on a relational structure,is the Boolean algebra with operators

C(X) = hF ;\;[; 0; ;; X; (fi)Ii

where for all i 2 I , if Ri is a relation of arity n+ 1 , then fi is an operator of arity n and for all S1; :::; Sn 2 F

fi(S1; :::; Sn) = fx 2 X : there exist x1 2 S1; :::; xn 2 Sn such that Ri(x1; :::; xn; x)g

This construction can be generalized to elds of sets on arbitrary algebraic structures having both operators andrelations as operators can be viewed as a special case of relations. If F is the whole power set of X then C(X) iscalled a full complex algebra or power algebra.

• 6 CHAPTER 2. FIELD OF SETS

Every (normal) Boolean algebra with operators can be represented as a eld of sets on a relational structure in thesense that it is isomorphic to the complex algebra corresponding to the eld.(Historically the term complex was rst used in the case where the algebraic structure was a group and has its originsin 19th century group theory where a subset of a group was called a complex.)

2.3 See also List of Boolean algebra topics Algebra of sets Sigma algebra Measure theory Probability theory Interior algebra Alexandrov topology Stones representation theorem for Boolean algebras Stone duality Boolean ring Preordered eld

2.4 References Goldblatt, R., Algebraic Polymodal Logic: A Survey, Logic Journal of the IGPL, Volume 8, Issue 4, p. 393-450,July 2000

Goldblatt, R., Varieties of complex algebras, Annals of Pure and Applied Logic, 44, p. 173-242, 1989 Johnstone, Peter T. (1982). Stone spaces (3rd ed.). Cambridge: Cambridge University Press. ISBN 0-521-33779-8.

Naturman, C.A., Interior Algebras and Topology, Ph.D. thesis, University of Cape Town Department of Math-ematics, 1991

Patrick Blackburn, Johan F.A.K. van Benthem, Frank Wolter ed., Handbook of Modal Logic, Volume 3 ofStudies in Logic and Practical Reasoning, Elsevier, 2006

2.5 External links Hazewinkel, Michiel, ed. (2001), Algebra of sets, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Chapter 3

Finite intersection property

In general topology, a branch of mathematics, a collectionA of subsets of a setX is said to have the nite intersectionproperty (FIP) if the intersection over any nite subcollection of A is nonempty. It has the strong nite intersectionproperty (SFIP) if the intersection over any nite subcollection of A is innite.A centered system of sets is a collection of sets with the nite intersection property.

3.1 DenitionLetX be a set withA = fAigi2I a family of subsets ofX . Then the collectionA has the nite intersection property(FIP), if any nite subcollection J I has non-empty intersectionTi2J Ai:3.2 DiscussionClearly the empty set cannot belong to any collection with the nite intersection property. The condition is triviallysatised if the intersection over the entire collection is nonempty (in particular, if the collection itself is empty), andit is also trivially satised if the collection is nested, meaning that the collection is totally ordered by inclusion (equiv-alently, for any nite subcollection, a particular element of the subcollection is contained in all the other elements ofthe subcollection), e.g. the nested sequence of intervals (0, 1/n). These are not the only possibilities however. Forexample, if X = (0, 1) and for each positive integer i, Xi is the set of elements of X having a decimal expansion withdigit 0 in the i'th decimal place, then any nite intersection is nonempty (just take 0 in those nitely many places and1 in the rest), but the intersection of all Xi for i 1 is empty, since no element of (0, 1) has all zero digits.The nite intersection property is useful in formulating an alternative denition of compactness: a space is compact ifand only if every collection of closed sets satisfying the nite intersection property has nonempty intersection itself.This formulation of compactness is used in some proofs of Tychonos theorem and the uncountability of the realnumbers (see next section)

3.3 ApplicationsTheorem. Let X be a non-empty compact Hausdor space that satises the property that no one-point set is open.Then X is uncountable.Proof. We will show that if U X is nonempty and open, and if x is a point of X, then there is a neighbourhoodV U whose closure doesnt contain x (x may or may not be in U). Choose y in U dierent from x (if x is in U,then there must exist such a y for otherwise U would be an open one point set; if x isnt in U, this is possible sinceU is nonempty). Then by the Hausdor condition, choose disjoint neighbourhoodsW and K of x and y respectively.Then K U will be a neighbourhood of y contained in U whose closure doesnt contain x as desired.

Now suppose f : N X is a bijection, and let {xi : i N} denote the image of f. Let X be the rst open set

7

• 8 CHAPTER 3. FINITE INTERSECTION PROPERTY

and choose a neighbourhood U1 X whose closure doesnt contain x1. Secondly, choose a neighbourhood U2 U1 whose closure doesnt contain x2. Continue this process whereby choosing a neighbourhood Un Un whoseclosure doesnt contain xn. Then the collection {Ui : i N} satises the nite intersection property and hence theintersection of their closures is nonempty (by the compactness of X). Therefore there is a point x in this intersection.No xi can belong to this intersection because xi doesnt belong to the closure of Ui. This means that x is not equal toxi for all i and f is not surjective; a contradiction. Therefore, X is uncountable.All the conditions in the statement of the theorem are necessary:1. We cannot eliminate the Hausdor condition; a countable set with the indiscrete topology is compact, has morethan one point, and satises the property that no one point sets are open, but is not uncountable.2. We cannot eliminate the compactness condition as the set of all rational numbers shows.3. We cannot eliminate the condition that one point sets cannot be open as a nite space as the discrete topologyshows.Corollary. Every closed interval [a, b] with a < b is uncountable. Therefore, R is uncountable.Corollary. Every perfect, locally compact Hausdor space is uncountable.Proof. Let X be a perfect, compact, Hausdor space, then the theorem immediately implies that X is uncountable.If X is a perfect, locally compact Hausdor space which is not compact, then the one-point compactication of X isa perfect, compact Hausdor space. Therefore the one point compactication of X is uncountable. Since removinga point from an uncountable set still leaves an uncountable set, X is uncountable as well.

3.4 ExamplesA lter has the nite intersection property by denition.

3.5 TheoremsLet X be nonempty, F 2X, F having the nite intersection property. Then there exists an F ultralter (in 2X) suchthat F F.See details and proof in Csirmaz & Hajnal (1994). This result is known as ultralter lemma.

3.6 VariantsA family of sets A has the strong nite intersection property (sp), if every nite subfamily of A has inniteintersection.

3.7 References A space is compact i any family of closed sets having p has non-empty intersection at PlanetMath.org.

 Csirmaz, Lszl; Hajnal, Andrs (1994), Matematikai logika (In Hungarian), Budapest: Etvs Lornd University.

Finite intersection property at PlanetMath.org.

• Chapter 4

Helly family

In combinatorics, a Helly family of order k is a family of sets such that any minimal subfamily with an emptyintersection has k or fewer sets in it. Equivalently, every nite subfamily such that every k -fold intersection is non-empty has non-empty total intersection.

The k-Helly property is the property of being a Helly family of order k. These concepts are named after EduardHelly (1884 - 1943); Hellys theorem on convex sets, which gave rise to this notion, states that convex sets in Euclideanspace of dimension n are a Helly family of order n + 1. The number k is frequently omitted from these names inthe case that k = 2.

4.1 Examples

In the family of all subsets of the set {a,b,c,d}, the subfamily {{a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}} has an emptyintersection, but removing any set from this subfamily causes it to have a nonempty intersection. Therefore,it is a minimal subfamily with an empty intersection. It has four sets in it, and is the largest possible minimalsubfamily with an empty intersection, so the family of all subsets of the set {a,b,c,d} is a Helly family of order4.

Let I be a nite set of closed intervals of the real line with an empty intersection. Let A be the interval whose leftendpoint a is as large as possible, and let B be the interval whose right endpoint b is as small as possible. Then,if a were less than or equal to b, all numbers in the range [a,b] would belong to all invervals of I, violatingthe assumption that the intersection of I is empty, so it must be the case that a > b. Thus, the two-intervalsubfamily {A,B} has an empty intersection, and the family I cannot be minimal unless I = {A,B}. Therefore,all minimal families of intervals with empty intersections have two or fewer intervals in them, showing that theset of all intervals is a Helly family of order 2.

The family of innite arithmetic progressions of integers also has the 2-Helly property. That is, whenever anite collection of progressions has the property that no two of them are disjoint, then there exists an integerthat belongs to all of them; this is the Chinese remainder theorem.

4.2 Formal denitionMore formally, a Helly family of order k is a set system (F, E), with F a collection of subsets of E, such that, forevery nite G F with

\X2G

X = ?;

we can nd H G such that

9

• 10 CHAPTER 4. HELLY FAMILY

\X2H

X = ?

and

jHj k: 

In some cases, the same denition holds for every subcollection G, regardless of niteness. However, this is a morerestrictive condition. For instance, the open intervals of the real line satisfy the Helly property for nite subcollections,but not for innite subcollections: the intervals (0,1/i) (for i = 0, 1, 2, ...) have pairwise nonempty intersections, buthave an empty overall intersection.

4.3 Helly dimensionIf a family of sets is a Helly family of order k, that family is said to have Helly number k. The Helly dimension ofa metric space is one less than the Helly number of the family of metric balls in that space; Hellys theorem impliesthat the Helly dimension of a Euclidean space equals its dimension as a real vector space.

The Helly dimension of a subset S of a Euclidean space, such as a polyhedron, is one less than the Helly number ofthe family of translates of S. For instance, the Helly dimension of any hypercube is 1, even though such a shapemay belong to a Euclidean space of much higher dimension.

Helly dimension has also been applied to other mathematical objects. For instance Domokos (2007) denes the Hellydimension of a group (an algebraic structure formed by an invertible and associative binary operation) to be one lessthan the Helly number of the family of left cosets of the group.

4.4 The Helly propertyIf a family of nonempty sets has an empty intersection, its Helly number must be at least two, so the smallest k forwhich the k-Helly property is nontrivial is k = 2. The 2-Helly property is also known as theHelly property. A 2-Hellyfamily is also known as a Helly family.

A convex metric space in which the closed balls have the 2-Helly property (that is, a space with Helly dimension 1, inthe stronger variant of Helly dimension for innite subcollections) is called injective or hyperconvex. The existenceof the tight span allows any metric space to be embedded isometrically into a space with Helly dimension 1.

4.5 References Bollobs, Bla (1986), Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability, Cam-

bridge University Press, p. 82, ISBN 9780521337038.

 Duchet, Pierre (1995), Hypergraphs, in Graham, R. L.; Grtschel, M.; Lovsz, L., Handbook of combinatorics, Vol. 1,2, Amsterdam: Elsevier, pp. 381432, MR 1373663. See in particular Section 2.5, Helly Property, pp. 393394.

 This is the one-dimensional case of Hellys theorem. For essentially this proof, with a colorful phrasing involving sleep-ing students, see Savchev, Svetoslav; Andreescu, Titu (2003), 27 Hellys Theorem for One Dimension, MathematicalMiniatures, New Mathematical Library 43, Mathematical Association of America, pp. 104106, ISBN 9780883856451.

 Martini, Horst (1997), Excursions Into Combinatorial Geometry, Springer, pp. 9293, ISBN 9783540613411.

 Bezdek, Kroly (2010), Classical Topics in Discrete Geometry, Springer, p. 27, ISBN 9781441906007.

 Sz.-Nagy, Bla (1954), Ein Satz ber Parallelverschiebungen konvexer Krper, Acta Universitatis Szegediensis 15: 169177, MR 0065942.

 Domokos,M. (2007), Typical separating invariants, TransformationGroups 12 (1): 4963, arXiv:math/0511300, doi:10.1007/s00031-005-1131-4, MR 2308028.

• 4.5. REFERENCES 11

 Deza, Michel Marie; Deza, Elena (2012), Encyclopedia of Distances, Springer, p. 19, ISBN 9783642309588

 Isbell, J. R. (1964), Six theorems about injectivemetric spaces, Comment. Math. Helv. 39: 6576, doi:10.1007/BF02566944.

• Chapter 5

Ring of sets

Not to be confused with Ring (mathematics).

In mathematics, there are two dierent notions of a ring of sets, both referring to certain families of sets. In ordertheory, a nonempty family of setsR is called a ring (of sets) if it is closed under intersection and union. That is, thefollowing two statements are true for all sets A and B ,

1. A;B 2 R implies A \B 2 R and2. A;B 2 R implies A [B 2 R: 

In measure theory, a ring of setsR is instead a nonempty family closed under unions and set-theoretic dierences.That is, the following two statements are true for all sets A and B (including when they are the same set),

1. A;B 2 R implies A nB 2 R and2. A;B 2 R implies A [B 2 R:

This implies the empty set is in R . It also implies that R is closed under symmetric dierence and intersection,because of the identities

1. A4B = (A nB) [ (B nA) and2. A \B = A n (A nB):

(So a ring in the second, measure theory, sense is also a ring in the rst, order theory, sense.) Together, theseoperations give R the structure of a boolean ring. Conversely, every family of sets closed under both symmetricdierence and intersection is also closed under union and dierences. This is due to the identities

1. A [B = (A4B)4 (A \B) and2. A nB = A4 (A \B):

5.1 ExamplesIf X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.If (X,) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongsto an upper set U and x y, then y must also belong to U) are closed under both intersections and unions. However,in general it will not be closed under dierences of sets.The open sets and closed sets of any topological space are closed under both unions and intersections.

12

• 5.2. RELATED STRUCTURES 13

On the real line R, the family of sets consisting of the empty set and all nite unions of intervals of the form (a, b],a,b in R is a ring in the measure theory sense.If T is any transformation dened on a space, then the sets that are mapped into themselves by T are closed underboth unions and intersections.

If two rings of sets are both dened on the same elements, then the sets that belong to both rings themselves form aring of sets.

5.2 Related structuresA ring of sets (in the order-theoretic sense) forms a distributive lattice in which the intersection and union operationscorrespond to the lattices meet and join operations, respectively. Conversely, every distributive lattice is isomorphicto a ring of sets; in the case of nite distributive lattices, this is Birkhos representation theorem and the sets maybe taken as the lower sets of a partially ordered set.

A eld of subsets of X is a ring that contains X and is closed under relative complement. Every eld, and so alsoevery -algebra, is a ring of sets in the measure theory sense.A semi-ring (of sets) is a family of sets S with the properties

1. ; 2 S;2. A;B 2 S implies A \B 2 S; and3. A;B 2 S implies A nB = Sni=1 Ci for some disjoint C1; : : : ; Cn 2 S:

Clearly, every ring (in the measure theory sense) is a semi-ring.A semi-eld of subsets of X is a semi-ring that contains X.

5.3 References Birkho, Garrett (1937), Rings of sets,DukeMathematical Journal 3 (3): 443454, doi:10.1215/S0012-7094-37-00334-

X, MR 1546000.

 De Barra, Gar (2003), Measure Theory and Integration, Horwood Publishing, p. 13, ISBN 9781904275046.

5.4 External links Ring of sets at Encyclopedia of Mathematics

• Chapter 6

Sigma-algebra

"-algebra redirects here. For an algebraic structure admitting a given signature of operations, see Universal al-gebra.

In mathematical analysis and in probability theory, a -algebra (also sigma-algebra, -eld, sigma-eld) on a setX is a collection of subsets of X that is closed under countable-fold set operations (complement, union of countablymany sets and intersection of countably many sets). By contrast, an algebra is only required to be closed under nitelymany set operations. That is, a -algebra is an algebra of sets, completed to include countably innite operations.The pair (X, ) is also a eld of sets, called a measurable space.The main use of -algebras is in the denition of measures; specically, the collection of those subsets for whicha given measure is dened is necessarily a -algebra. This concept is important in mathematical analysis as thefoundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events whichcan be assigned probabilities. Also, in probability, -algebras are pivotal in the denition of conditional expectation.In statistics, (sub) -algebras are needed for a formal mathematical denition of sucient statistic, particularlywhen the statistic is a function or a random process and the notion of conditional density is not applicable.If X = {a, b, c, d}, one possible -algebra on X is = { , {a, b}, {c, d}, {a, b, c, d} }, where is the empty set.However, a nite algebra is always a -algebra.If {A1, A2, A3, } is a countable partition of X then the collection of all unions of sets in the partition (includingthe empty set) is a -algebra.A more useful example is the set of subsets of the real line formed by starting with all open intervals and addingin all countable unions, countable intersections, and relative complements and continuing this process (by transniteiteration through all countable ordinals) until the relevant closure properties are achieved (a construction known asthe Borel hierarchy).

6.1 MotivationThere are at least three key motivators for -algebras: dening measures, manipulating limits of sets, and managingpartial information characterized by sets.

6.1.1 Measure

Ameasure on X is a function that assigns a non-negative real number to subsets of X; this can be thought of as makingprecise a notion of size or volume for sets. We want the size of the union of disjoint sets to be the sum of theirindividual sizes, even for an innite sequence of disjoint sets.One would like to assign a size to every subset of X, but in many natural settings, this is not possible. For example theaxiom of choice implies that when the size under consideration is the ordinary notion of length for subsets of the realline, then there exist sets for which no size exists, for example, the Vitali sets. For this reason, one considers instead asmaller collection of privileged subsets of X. These subsets will be called the measurable sets. They are closed under

14

• 6.1. MOTIVATION 15

operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable setand the countable union of measurable sets is a measurable set. Non-empty collections of sets with these propertiesare called -algebras.

6.1.2 Limits of setsMany uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences ofsets. For this, closure under countable unions and intersections is paramount. Set limits are dened as follows on-algebras.

The limit supremum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

lim supn!1

An =1\n=1

1[m=n

Am:

The limit inmum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

lim infn!1 An =

1[n=1

1\m=n

Am:

If, in fact,

lim infn!1 An = lim supn!1

An

then the limn!1An exists as that common set.

6.1.3 Sub -algebrasIn much of probability, especially when conditional expectation is involved, one is concerned with sets that representonly part of all the possible information that can be observed. This partial information can be characterized with asmaller -algebra which is a subset of the principal -algebra; it consists of the collection of subsets relevant only toand determined only by the partial information. A simple example suces to illustrate this idea.Imagine you are playing a game that involves ipping a coin repeatedly and observing whether it comes up Heads (H)or Tails (T). Since you and your opponent are each innitely wealthy, there is no limit to how long the game can last.This means the sample space must consist of all possible innite sequences of H or T :

= fH;Tg1 = f(x1; x2; x3; : : : ) : xi 2 fH;Tg; i 1gHowever, after n ips of the coin, you may want to determine or revise your betting strategy in advance of the nextip. The observed information at that point can be described in terms of the 2n possibilities for the rst n ips.Formally, since you need to use subsets of , this is codied as the -algebra

Gn = fA fH;Tg1 : A fH;TgngObserve that then

G1 G2 G3 G1where G1 is the smallest -algebra containing all the others.

• 16 CHAPTER 6. SIGMA-ALGEBRA

6.2 Denition and properties

6.2.1 DenitionLet X be some set, and let 2X represent its power set. Then a subset 2X is called a -algebra if it satises thefollowing three properties:

1. X is in , and X is considered to be the universal set in the following context.

2. is closed under complementation: If A is in , then so is its complement, X\A.

3. is closed under countable unions: If A1, A2, A3, ... are in , then so is A = A1 A2 A3 .

From these properties, it follows that the -algebra is also closed under countable intersections (by applying DeMorgans laws).It also follows that the empty set is in , since by (1) X is in and (2) asserts that its complement, the empty set,is also in . Moreover, by (3) it follows as well that {X, } is the smallest possible -algebra.Elements of the -algebra are called measurable sets. An ordered pair (X, ), where X is a set and is a -algebraover X, is called a measurable space. A function between two measurable spaces is called a measurable function ifthe preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with themeasurable functions as morphisms. Measures are dened as certain types of functions from a -algebra to [0, ].A -algebra is both a -system and a Dynkin system (-system). The converse is true as well, by Dynkins theorem(below).

6.2.2 Dynkins - theoremThis theorem (or the related monotone class theorem) is an essential tool for proving many results about propertiesof specic -algebras. It capitalizes on the nature of two simpler classes of sets, namely the following.

A -system P is a collection of subsets of that is closed under nitely many intersections, anda Dynkin system (or -system) D is a collection of subsets of that contains and is closed undercomplement and under countable unions of disjoint subsets.

Dynkins - theorem says, if P is a -system and D is a Dynkin system that contains P then the -algebra (P)generated by P is contained in D. Since certain -systems are relatively simple classes, it may not be hard to verifythat all sets in P enjoy the property under consideration while, on the other hand, showing that the collection D of allsubsets with the property is a Dynkin system can also be straightforward. Dynkins - Theorem then implies thatall sets in (P) enjoy the property, avoiding the task of checking it for an arbitrary set in (P).One of the most fundamental uses of the - theorem is to show equivalence of separately dened measures orintegrals. For example, it is used to equate a probability for a random variable X with the Lebesgue-Stieltjes integraltypically associated with computing the probability:

P(X 2 A) = RAF (dx) for all A in the Borel -algebra on R,

where F(x) is the cumulative distribution function for X, dened on R, while P is a probability measure, dened ona -algebra of subsets of some sample space .

6.2.3 Combining -algebrasSuppose f : 2 Ag is a collection of -algebras on a space X.

The intersection of a collection of -algebras is a -algebra. To emphasize its character as a -algebra, it oftenis denoted by:

• 6.2. DEFINITION AND PROPERTIES 17

^2A

:

Sketch of Proof: Let denote the intersection. Since X is in every , is not empty. Closure undercomplement and countable unions for every implies the same must be true for . Therefore, is a-algebra.

The union of a collection of -algebras is not generally a -algebra, or even an algebra, but it generates a-algebra known as the join which typically is denoted

_2A

=

[2A

!:

A -system that generates the join is

P =(

n\i=1

Ai : Ai 2 i ; i 2 A; n 1):

Sketch of Proof: By the case n = 1, it is seen that each P , so[2A

P:

This implies

[2A

! (P)

by the denition of a -algebra generated by a collection of subsets. On the other hand,

P [2A

!

which, by Dynkins - theorem, implies

(P) [2A

!:

6.2.4 -algebras for subspacesSuppose Y is a subset of X and let (X, ) be a measurable space.

The collection {Y B: B } is a -algebra of subsets of Y. Suppose (Y, ) is a measurable space. The collection {A X : A Y } is a -algebra of subsets of X.

6.2.5 Relation to -ringA -algebra is just a -ring that contains the universal set X. A -ring need not be a -algebra, as for examplemeasurable subsets of zero Lebesgue measure in the real line are a -ring, but not a -algebra since the real linehas innite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takesmeasurable subsets of nite Lebesgue measure, those are a ring but not a -ring, since the real line can be obtainedby their countable union yet its measure is not nite.

• 18 CHAPTER 6. SIGMA-ALGEBRA

6.2.6 Typographic note-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface. Thus (X, ) may bedenoted as (X;F) or (X;F) .

6.3 Examples

6.3.1 Simple set-based examplesLet X be any set.

The family consisting only of the empty set and the set X, called the minimal or trivial -algebra over X. The power set of X, called the discrete -algebra. The collection {, A, Ac, X} is a simple -algebra generated by the subset A. The collection of subsets of X which are countable or whose complements are countable is a -algebra (whichis distinct from the power set of X if and only if X is uncountable). This is the -algebra generated by thesingletons of X. Note: countable includes nite or empty.

The collection of all unions of sets in a countable partition of X is a -algebra.

6.3.2 Stopping time sigma-algebrasA stopping time can dene a -algebraF , the so-called stopping time sigma-algebra, which in a ltered probabilityspace describes the information up to the random time in the sense that, if the ltered probability space is interpretedas a random experiment, the maximum information that can be found out about the experiment from arbitrarily oftenrepeating it until the time is F .

6.4 -algebras generated by families of sets

6.4.1 -algebra generated by an arbitrary familyLet F be an arbitrary family of subsets of X. Then there exists a unique smallest -algebra which contains every setin F (even though F may or may not itself be a -algebra). It is, in fact, the intersection of all -algebras containingF. (See intersections of -algebras above.) This -algebra is denoted (F) and is called the -algebra generated byF.For a simple example, consider the set X = {1, 2, 3}. Then the -algebra generated by the single subset {1} is({{1}}) = {, {1}, {2, 3}, {1, 2, 3}}. By an abuse of notation, when a collection of subsets contains only oneelement, A, one may write (A) instead of ({A}); in the prior example ({1}) instead of ({{1}}). Indeed, using(A1, A2, ...) to mean ({A1, A2, ...}) is also quite common.There are many families of subsets that generate useful -algebras. Some of these are presented here.

6.4.2 -algebra generated by a functionIf f is a function from a set X to a set Y and B is a -algebra of subsets of Y, then the -algebra generated by thefunction f, denoted by (f), is the collection of all inverse images f1(S) of the sets S in B. i.e.

(f) = ff1(S) jS 2 Bg:A function f from a set X to a set Y is measurable with respect to a -algebra of subsets of X if and only if (f) isa subset of .

• 6.4. -ALGEBRAS GENERATED BY FAMILIES OF SETS 19

One common situation, and understood by default if B is not specied explicitly, is when Y is a metric or topologicalspace and B is the collection of Borel sets on Y.If f is a function from X to Rn then (f) is generated by the family of subsets which are inverse images of inter-vals/rectangles in Rn:

(f) = ff1((a1; b1] (an; bn]) : ai; bi 2 Rg :

A useful property is the following. Assume f is a measurable map from (X, X) to (S, S) and g is a measurable mapfrom (X, X) to (T, T). If there exists a measurable function h from T to S such that f(x) = h(g(x)) then (f) (g). If S is nite or countably innite or if (S, S) is a standard Borel space (e.g., a separable complete metric spacewith its associated Borel sets) then the converse is also true. Examples of standard Borel spaces include Rn with itsBorel sets and R with the cylinder -algebra described below.

6.4.3 Borel and Lebesgue -algebrasAn important example is the Borel algebra over any topological space: the -algebra generated by the open sets (or,equivalently, by the closed sets). Note that this -algebra is not, in general, the whole power set. For a non-trivialexample that is not a Borel set, see the Vitali set or Non-Borel sets.On the Euclidean space Rn, another -algebra is of importance: that of all Lebesgue measurable sets. This -algebracontains more sets than the Borel -algebra onRn and is preferred in integration theory, as it gives a complete measurespace.

6.4.4 Product -algebraLet (X1;1) and (X2;2) be two measurable spaces. The -algebra for the corresponding product spaceX1 X2is called the product -algebra and is dened by

1 2 = (fB1 B2 : B1 2 1; B2 2 2g):Observe that fB1 B2 : B1 2 1; B2 2 2g is a -system.The Borel -algebra for Rn is generated by half-innite rectangles and by nite rectangles. For example,

B(Rn) = (f(1; b1] (1; bn] : bi 2 Rg) = (f(a1; b1] (an; bn] : ai; bi 2 Rg) :For each of these two examples, the generating family is a -system.

6.4.5 -algebra generated by cylinder setsSuppose

X RT = ff : f(t) 2 R; t 2 Tgis a set of real-valued functions. Let B(R) denote the Borel subsets ofR. A cylinder subset of X is a nitely restrictedset dened as

Ct1;:::;tn(B1; : : : ; Bn) = ff 2 X : f(ti) 2 Bi; 1 i ng:Each

fCt1;:::;tn(B1; : : : ; Bn) : Bi 2 B(R); 1 i ng

• 20 CHAPTER 6. SIGMA-ALGEBRA

is a -system that generates a -algebra t1;:::;tn . Then the family of subsets

FX =1[n=1

[ti2T;in

t1;:::;tn

is an algebra that generates the cylinder -algebra for X. This -algebra is a subalgebra of the Borel -algebradetermined by the product topology of RT restricted to X.An important special case is when T is the set of natural numbers and X is a set of real-valued sequences. In thiscase, it suces to consider the cylinder sets

Cn(B1; : : : ; Bn) = (B1 Bn R1) \X = f(x1; x2; : : : ; xn; xn+1; : : : ) 2 X : xi 2 Bi; 1 i ng;

for which

n = (fCn(B1; : : : ; Bn) : Bi 2 B(R); 1 i ng)

is a non-decreasing sequence of -algebras.

6.4.6 -algebra generated by random variable or vector

Suppose (;;P) is a probability space. If Y : ! Rn is measurable with respect to the Borel -algebra on Rnthen Y is called a random variable (n = 1) or random vector (n 1). The -algebra generated by Y is

(Y ) = fY 1(A) : A 2 B(Rn)g:

6.4.7 -algebra generated by a stochastic process

Suppose (;;P) is a probability space and RT is the set of real-valued functions on T . If Y : ! X RT ismeasurable with respect to the cylinder -algebra (FX) (see above) for X then Y is called a stochastic process orrandom process. The -algebra generated by Y is

(Y ) =Y 1(A) : A 2 (FX)

= (fY 1(A) : A 2 FXg);

the -algebra generated by the inverse images of cylinder sets.

Measurable function

Sample space

Separable sigma algebra

Sigma ring

• 6.6. REFERENCES 21

6.6 References Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Wiley. ISBN 978-1-118-12237-2.

 Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.

 Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiley & Sons. p. 12. ISBN 978-0-470-31795-2.

 Fischer, Tom (2013). On simple representations of stopping times and stopping time sigma-algebras. Statistics andProbability Letters 83 (1): 345349. doi:10.1016/j.spl.2012.09.024.

 Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Springer. p. 7. ISBN 0-387-95313-2.

6.7 External links Hazewinkel, Michiel, ed. (2001), Algebra of sets, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Sigma Algebra from PlanetMath.

• Chapter 7

Sigma-ideal

In mathematics, particularly measure theory, a -ideal of a sigma-algebra (, read sigma, means countable in thiscontext) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent applicationis perhaps in probability theory.Let (X,) be a measurable space (meaning is a -algebra of subsets of X). A subset N of is a -ideal if thefollowing properties are satised:(i) N;(ii) When A N and B , B A B N;(iii) fAngn2N N )

Sn2NAn 2 N:

Briey, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. Theconcept of -ideal is dual to that of a countably complete (-) lter.If a measure is given on (X,), the set of -negligible sets (S such that (S) = 0) is a -ideal.The notion can be generalized to preorders (P,,0) with a bottom element 0 as follows: I is a -ideal of P just when(i') 0 I,(ii') x y & y I x I, and(iii') given a family xn I (n N), there is y I such that xn y for each nThus I contains the bottom element, is downward closed, and is closed under countable suprema (which must exist).It is natural in this context to ask that P itself have countable suprema.

7.1 References Bauer, Heinz (2001): Measure and Integration Theory. Walter de Gruyter GmbH & Co. KG, 10785 Berlin,Germany.

22

• Chapter 8

Sigma-ring

Inmathematics, a nonempty collection of sets is called a -ring (pronounced sigma-ring) if it is closed under countableunion and relative complementation.

8.1 Formal denitionLetR be a nonempty collection of sets. ThenR is a -ring if:

1. S1n=1An 2 R if An 2 R for all n 2 N2. A nB 2 R if A;B 2 R

8.2 PropertiesFrom these two properties we immediately see that

T1n=1An 2 R if An 2 R for all n 2 N

This is simply because \1n=1An = A1 n [1n=1(A1 nAn) .

8.3 Similar conceptsIf the rst property is weakened to closure under nite union (i.e.,A[B 2 RwheneverA;B 2 R ) but not countableunion, thenR is a ring but not a -ring.

8.4 Uses-rings can be used instead of -elds (-algebras) in the development of measure and integration theory, if one doesnot wish to require that the universal set be measurable. Every -eld is also a -ring, but a -ring need not be a-eld.A -ring R that is a collection of subsets of X induces a -eld for X . Dene A to be the collection of all subsetsof X that are elements ofR or whose complements are elements ofR . Then A is a -eld over the set X . In factA is the minimal -eld containingR since it must be contained in every -eld containingR .

23

• 24 CHAPTER 8. SIGMA-RING

8.6 References Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses -ringsin development of Lebesgue theory.

• Chapter 9

Steiner system

The Fano plane is an S(2,3,7) Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongsto a unique line.

In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specically at-design with = 1 and t 2.

25

• 26 CHAPTER 9. STEINER SYSTEM

A Steiner system with parameters t, k, n, written S(t,k,n), is an n-element set S together with a set of k-element subsetsof S (called blocks) with the property that each t-element subset of S is contained in exactly one block. In an alternatenotation for block designs, an S(t,k,n) would be a t-(n,k,1) design.This denition is relatively modern, generalizing the classical denition of Steiner systems which in addition requiredthat k = t + 1. An S(2,3,n) was (and still is) called a Steiner triple (or triad) system, while an S(3,4,n) was called aSteiner quadruple system, and so on. With the generalization of the denition, this naming system is no longer strictlyadhered to.A long-standing problem in design theory is if any nontrivial (t < k < n) Steiner systems have t 6; also if innitelymany have t = 4 or 5. This was claimed to be solved in the armative by Peter Keevash.

9.1 Examples

9.1.1 Finite projective planesA nite projective plane of order q, with the lines as blocks, is an S(2; q + 1; q2 + q + 1) , since it has q2 + q + 1points, each line passes through q + 1 points, and each pair of distinct points lies on exactly one line.

9.1.2 Finite ane planesA nite ane plane of order q, with the lines as blocks, is an S(2, q, q2). An ane plane of order q can be obtainedfrom a projective plane of the same order by removing one block and all of the points in that block from the projectiveplane. Choosing dierent blocks to remove in this way can lead to non-isomorphic ane planes.

9.2 Classical Steiner systems

9.2.1 Steiner triple systemsAn S(2,3,n) is called a Steiner triple system, and its blocks are called triples. It is common to see the abbreviationSTS(n) for a Steiner triple system of order n. The number of triples is n(n1)/6. A necessary and sucient conditionfor the existence of an S(2,3,n) is that n 1 or 3 (mod 6). The projective plane of order 2 (the Fano plane) is anSTS(7) and the ane plane of order 3 is an STS(9).Up to isomorphism, the STS(7) and STS(9) are unique, there are two STS(13)s, 80 STS(15)s, and 11,084,874,829STS(19)s.

We can dene a multiplication on the set S using the Steiner triple system by setting aa = a for all a in S, and ab= c if {a,b,c} is a triple. This makes S an idempotent, commutative quasigroup. It has the additional property thatab = c implies bc = a and ca = b. Conversely, any (nite) quasigroup with these properties arises froma Steiner triple system. Commutative idempotent quasigroups satisfying this additional property are called Steinerquasigroups.

9.2.2 Steiner quadruple systemsAn S(3,4,n) is called a Steiner quadruple system. A necessary and sucient condition for the existence of anS(3,4,n) is that n 2 or 4 (mod 6). The abbreviation SQS(n) is often used for these systems.Up to isomorphism, SQS(8) and SQS(10) are unique, there are 4 SQS(14)s and 1,054,163 SQS(16)s.

9.2.3 Steiner quintuple systemsAn S(4,5,n) is called a Steiner quintuple system. A necessary condition for the existence of such a system is that n 3or 5 (mod 6) which comes from considerations that apply to all the classical Steiner systems. An additional necessarycondition is that n 6 4 (mod 5), which comes from the fact that the number of blocks must be an integer. Sucientconditions are not known.

• 9.3. PROPERTIES 27

There is a unique Steiner quintuple system of order 11, but none of order 15 or order 17. Systems are known fororders 23, 35, 47, 71, 83, 107, 131, 167 and 243. The smallest order for which the existence is not known (as of2011) is 21.

9.3 PropertiesIt is clear from the denition of S(t,k,n) that 1 < t < k < n . (Equalities, while technically possible, lead to trivialsystems.)If S(t,k,n) exists, then taking all blocks containing a specic element and discarding that element gives a derived systemS(t1,k1,n1). Therefore the existence of S(t1,k1,n1) is a necessary condition for the existence of S(t,k,n).The number of t-element subsets in S is

nt

, while the number of t-element subsets in each block is

kt

. Since

every t-element subset is contained in exactly one block, we havent

= bkt

, or b =

nt

kt

, where b is the numberof blocks. Similar reasoning about t-element subsets containing a particular element gives us

n1t1= r

k1t1,

or r =n1t1

k1t1 , where r is the number of blocks containing any given element. From these denitions follows the

equation bk = rn . It is a necessary condition for the existence of S(t,k,n) that b and r are integers. As with anyblock design, Fishers inequality b n is true in Steiner systems.Given the parameters of a Steiner system S(t,k,n) and a subset of size t0 t , contained in at least one block, onecan compute the number of blocks intersecting that subset in a xed number of elements by constructing a Pascaltriangle. In particular, the number of blocks intersecting a xed block in any number of elements is independent ofthe chosen block.It can be shown that if there is a Steiner system S(2,k,n), where k is a prime power greater than 1, then n 1 or k(mod k(k1)). In particular, a Steiner triple system S(2,3,n) must have n = 6m+1 or 6m+3. It is known that this is theonly restriction on Steiner triple systems, that is, for each natural number m, systems S(2,3,6m+1) and S(2,3,6m+3)exist.

9.4 HistorySteiner triple systems were dened for the rst time by W.S.B. Woolhouse in 1844 in the Prize question #1733 ofLadys and Gentlemens Diary. The posed problem was solved by Thomas Kirkman (1847). In 1850 Kirkmanposed a variation of the problem known as Kirkmans schoolgirl problem, which asks for triple systems having anadditional property (resolvability). Unaware of Kirkmans work, Jakob Steiner (1853) reintroduced triple systems,and as this work was more widely known, the systems were named in his honor.

9.5 Mathieu groupsSeveral examples of Steiner systems are closely related to group theory. In particular, the nite simple groups calledMathieu groups arise as automorphism groups of Steiner systems:

The Mathieu group M11 is the automorphism group of a S(4,5,11) Steiner system

The Mathieu group M12 is the automorphism group of a S(5,6,12) Steiner system

The Mathieu group M22 is the unique index 2 subgroup of the automorphism group of a S(3,6,22) Steinersystem

The Mathieu group M23 is the automorphism group of a S(4,7,23) Steiner system

The Mathieu group M24 is the automorphism group of a S(5,8,24) Steiner system.

• 28 CHAPTER 9. STEINER SYSTEM

9.6 The Steiner system S(5, 6, 12)There is a unique S(5,6,12) Steiner system; its automorphism group is the Mathieu group M12, and in that context itis denoted by W12.

9.6.1 Constructions

There are dierent ways to construct an S(5,6,12) system.

Projective line method

This construction is due to Carmichael (1937).

Add a new element, call it , to the 11 elements of the nite eld F11 (that is, the integers mod 11). This set, S,of 12 elements can be formally identied with the points of the projective line over F11. Call the following specicsubset of size 6,

f1; 1; 3; 4; 5; 9g;

a block. From this block, we obtain the other blocks of the S(5,6,12) system by repeatedly applying the linearfractional transformations:

z0 = f(z) =az + b

cz + dwhere ,a; b; c; d in are F11 and ad bc in square non-zero a is F11:

With the usual conventions of dening f (d/c) = and f () = a/c, these functions map the set S onto itself. Ingeometric language, they are projectivities of the projective line. They form a group under composition which is theprojective special linear group PSL(2,11) of order 660. There are exactly ve elements of this group that leave thestarting block xed setwise, so there will be 132 images of that block. As a consequence of the multiply transitiveproperty of this group acting on this set, any subset of ve elements of S will appear in exactly one of these 132images of size six.

Kitten method

An alternative construction of W12 is obtained by use of the 'kitten' of R.T. Curtis, which was intended as a handcalculator to write down blocks one at a time. The kitten method is based on completing patterns in a 3x3 grid ofnumbers, which represent an ane geometry on the vector space F3xF3, an S(2,3,9) system.

Construction from K6 graph factorization

The relations between the graph factors of the complete graph K6 generate an S(5,6,12). AK6 graph has 6 dierent1-factorizations (ways to partition the edges into disjoint perfect matchings), and also 6 vertices. The set of verticesand the set of factorizations provide one block each. For every distinct pair of factorizations, there exists exactly oneperfect matching in common. Take the set of vertices and replace the two vertices corresponding to an edge of thecommon perfect matching with the labels corresponding to the factorizations; add that to the set of blocks. Repeatthis with the other two edges of the common perfect matching. Similarly take the set of factorizations and replacethe labels corresponding to the two factorizations with the end points of an edge in the common perfect matching.Repeat with the other two edges in the matching. There are thus 3+3 = 6 blocks per pair of factorizations, and thereare 6C2 = 15 pairs among the 6 factorizations, resulting in 90 new blocks. Finally take the full set of 12C6 = 924combinations of 6 objects out of 12, and discard any combination that has 5 or more objects in common with any ofthe 92 blocks generated so far. Exactly 40 blocks remain, resulting in 2+90+40 = 132 blocks of the S(5,6,12).

• 9.7. THE STEINER SYSTEM S(5, 8, 24) 29

9.7 The Steiner system S(5, 8, 24)The Steiner system S(5, 8, 24), also known as theWitt design orWitt geometry, was rst described by Carmichael(1931) and rediscovered by Witt (1938). This system is connected with many of the sporadic simple groups and withthe exceptional 24-dimensional lattice known as the Leech lattice.The automorphism group of S(5, 8, 24) is the Mathieu group M24, and in that context the design is denoted W24(W for Witt)

9.7.1 ConstructionsThere are many ways to construct the S(5,8,24). Two methods are described here:

Method based on 8-combinations of 24 elements

All 8-element subsets of a 24-element set are generated in lexicographic order, and any such subset which diersfrom some subset already found in fewer than four positions is discarded.The list of octads for the elements 01, 02, 03, ..., 22, 23, 24 is then:

01 02 03 04 05 06 07 0801 02 03 04 09 10 11 1201 02 03 04 13 14 15 16.. (next 753 octads omitted).13 14 15 16 17 18 19 2013 14 15 16 21 22 23 2417 18 19 20 21 22 23 24

Method based on 24-bit binary strings

All 24-bit binary strings are generated in lexicographic order, and any such string that diers from some earlier onein fewer than 8 positions is discarded. The result looks like this:000000000000000000000000 000000000000000011111111 000000000000111100001111 000000000000111111110000000000000011001100110011 000000000011001111001100 000000000011110000111100 000000000011110011000011000000000101010101010101 000000000101010110101010 . . (next 4083 24-bit strings omitted) . 111111111111000011110000111111111111111100000000 111111111111111111111111The list contains 4096 items, which are each code words of the extended binary Golay code. They form a groupunder the XOR operation. One of them has zero 1-bits, 759 of them have eight 1-bits, 2576 of them have twelve1-bits, 759 of them have sixteen 1-bits, and one has twenty-four 1-bits. The 759 8-element blocks of the S(5,8,24)(called octads) are given by the patterns of 1s in the code words with eight 1-bits.

• 30 CHAPTER 9. STEINER SYSTEM

9.9 Notes Encyclopaedia of Design Theory: t-Designs. Designtheory.org. 2004-10-04. Retrieved 2012-08-17. Keevash, Peter (2014). The existence of designs. arXiv:1401.3665. A Design Dilemma Solved, Minus Designs. Quanta Magazine. 2015-06-09. Retrieved 2015-06-27. Colbourn & Dinitz 2007, pg.60 This property is equivalent to saying that (xy)y = x for all x and y in the idempotent commutative quasigroup. Colbourn & Dinitz 2007, pg. 497, denition 28.12 Colbourn & Dinitz 2007, pg.106 stergard & Pottonen 2008 Assmus & Key 1994, pg. 8 Lindner & Rodger 1997, pg.3 Carmichael 1956, p. 431 Beth, Jungnickel & Lenz 1986, p. 196 Curtis 1984 EAGTS textbook

9.10 References Assmus, E. F., Jr.; Key, J. D. (1994), 8. Steiner Systems, Designs and Their Codes, Cambridge UniversityPress, pp. 295316, ISBN 0-521-45839-0.

Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge: Cambridge UniversityPress. 2nd ed. (1999) ISBN 978-0-521-44432-3.

Carmichael, Robert (1931), Tactical Congurations of Rank Two, American Journal of Mathematics 53:217240, doi:10.2307/2370885

Carmichael, Robert D. (1956) , Introduction to the theory of Groups of Finite Order, Dover, ISBN 0-486-60300-8

Colbourn, Charles J.; Dinitz, Jerey H. (1996), Handbook of Combinatorial Designs, Boca Raton: Chapman& Hall/ CRC, ISBN 0-8493-8948-8, Zbl 0836.00010

Colbourn, Charles J.; Dinitz, Jerey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:Chapman & Hall/ CRC, ISBN 1-58488-506-8, Zbl 1101.05001

Curtis, R.T. (1984), The Steiner system S(5,6,12), the Mathieu group M12 and the kitten"", in Atkinson,Michael D., Computational group theory (Durham, 1982), London: Academic Press, pp. 353358, ISBN0-12-066270-1, MR 0760669

Hughes, D. R.; Piper, F. C. (1985), Design Theory, Cambridge University Press, pp. 173176, ISBN 0-521-35872-8.

Kirkman, Thomas P. (1847), On a Problem in Combinations, The Cambridge and Dublin MathematicalJournal (Macmillan, Barclay, and Macmillan) II: 191204.

Lindner, C.C.; Rodger, C.A. (1997), Design Theory, Boca Raton: CRC Press, ISBN 0-8493-3986-3 stergard, Patric R.J.; Pottonen, Olli (2008), There exists no Steiner system S(4,5,17)", Journal of Combina-torial Theory Series A 115 (8): 15701573, doi:10.1016/j.jcta.2008.04.005

Steiner, J. (1853), Combinatorische Aufgabe, Journal fr die Reine und Angewandte Mathematik 45: 181182.

Witt, Ernst (1938), Die 5-Fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Univ. Hamburg 12:256264, doi:10.1007/BF02948947

9.11 External links Rowland, Todd and Weisstein, Eric W., Steiner System, MathWorld. Rumov, B.T. (2001), Steiner system, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

Steiner systems by Andries E. Brouwer Implementation of S(5,8,24) by Dr. Alberto Delgado, Gabe Hart, and Michael Kolkebeck S(5, 8, 24) Software and Listing by Johan E. Mebius The Witt Design computed by Ashay Dharwadker

• 32 CHAPTER 9. STEINER SYSTEM

9.12 Text and image sources, contributors, and licenses9.12.1 Text

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admissible setsAlexandrov representationalgebraalgebraalgebra of complexesalgebra overalgebraicalgebraicAlgebraic and canonical preorder fieldsAlgebraic fields of sets and Stone fieldsblockscanonicalcentered system of setscompactcomplexcomplexcomplexcomplexesConstruction from KCorollarycylinder -algebraDelta-ringSee also References

descriptivedifferentiateddiscrete -algebraeventsField of setsFields of sets in the representation theory of Boolean algebras Stone representation Separative and compact fields of sets towards Stone duality

Fields of sets with additional structure Sigma algebras and measure spaces Topological fields of sets Preorder fields Complex algebras and fields of sets on relational structures

field of setsfinite intersectionFinite intersection propertyDefinitionDiscussionApplicationsExamplesTheorems VariantsReferences

full complex algebrafunctiongraph factorizationHelly dimensionHelly dimensionHelly familyHelly familyExamples Formal definition Helly dimension The Helly property References

Helly family of orderHelly family of orderHelly numberHelly propertyHelly propertyidealKitten methodmeasurable setsmeasurable spacemeasurable spacemeasurable spacemeasure spaceMethod based on 8-combinations of 24 elementsMethod based on 24-bit binary stringspointspower algebrapower setpreorder fieldprobability spaceproduct -algebraProjective line methodProofProofpropertyRrandom processrandom variablerandom vectorrepresentationring of setsRing of setsExamplesRelated structuresReferencesExternal links

ring of setssamplesample spacesamplesseparativeSigma-algebraMotivationMeasureLimits of setsSub -algebras

Definition and propertiesDefinitionDynkins - theoremCombining -algebras-algebras for subspacesRelation to -ringTypographic note

ExamplesSimple set-based examplesStopping time sigma-algebras

-algebras generated by families of sets-algebra generated by an arbitrary family-algebra generated by a functionBorel and Lebesgue -algebrasProduct -algebra-algebra generated by cylinder sets-algebra generated by random variable or vector-algebra generated by a stochastic process

sigma-algebra -field sigma-fieldSigma-idealReferences